| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 6·13-s + 2·14-s + 16-s − 6·23-s − 6·26-s − 2·28-s + 6·29-s − 10·31-s − 32-s + 2·37-s + 4·43-s + 6·46-s + 8·47-s − 3·49-s + 6·52-s + 6·53-s + 2·56-s − 6·58-s − 10·61-s + 10·62-s + 64-s − 8·67-s + 10·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 1.66·13-s + 0.534·14-s + 1/4·16-s − 1.25·23-s − 1.17·26-s − 0.377·28-s + 1.11·29-s − 1.79·31-s − 0.176·32-s + 0.328·37-s + 0.609·43-s + 0.884·46-s + 1.16·47-s − 3/7·49-s + 0.832·52-s + 0.824·53-s + 0.267·56-s − 0.787·58-s − 1.28·61-s + 1.27·62-s + 1/8·64-s − 0.977·67-s + 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.343307410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.343307410\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.43318770322941, −12.97163016085400, −12.54172664526955, −11.90532836332029, −11.60136409307957, −10.83663134589732, −10.60902343056782, −10.16495708803180, −9.504868421475041, −9.073471210618694, −8.696493846709044, −8.199241520018136, −7.525951800455519, −7.244954743248016, −6.347965441947103, −6.160051226586049, −5.773766360006522, −4.948579078491381, −4.153275032381913, −3.668626847060905, −3.223491633353969, −2.455297345082855, −1.824066210132717, −1.132651147290383, −0.4288939469902108,
0.4288939469902108, 1.132651147290383, 1.824066210132717, 2.455297345082855, 3.223491633353969, 3.668626847060905, 4.153275032381913, 4.948579078491381, 5.773766360006522, 6.160051226586049, 6.347965441947103, 7.244954743248016, 7.525951800455519, 8.199241520018136, 8.696493846709044, 9.073471210618694, 9.504868421475041, 10.16495708803180, 10.60902343056782, 10.83663134589732, 11.60136409307957, 11.90532836332029, 12.54172664526955, 12.97163016085400, 13.43318770322941
